A Representation Theorem for Utility Maximization

A Representation Theorem for UtilityMaximization

Mark Dean

G4840 - Behavioral Economics

A Representation Theorem

• When dealing with models that have latent (or unobservable)variables we will want to find a representation theorem

• This consists of three things• A data set• A model• A set of conditions on the data which are necessary andsuffi cient for it to be consistent with the model

• Means testing these conditions is the same as testing themodel itself

A Representation Theorem for Utility Maximization

• We are now going to develop a representation theorem for themodel of utility maximization

• We want to do so properly, so we are going to have to usesome notation

• Don’t worry - we are just formalizing the ideas from before!

Data

• The data we are going to use are the choices people make• Notation:

• X : Set of objects you might get to choose from• 2X : The power set of X (i.e. all the subsets of X )• ∅: The empty set

• Our data is going to take the form of a choicecorrespondence which tells us what the person chose fromeach subset of X

DefinitionA choice correspondence C is a mapping C : 2X /∅→ 2X /∅ suchthat C (A) ⊂ A for all A ∈ 2X /∅.

Notes

• Don’t panic! This is just a way of recording what wedescribed previously

• For example, if we offered someone the choice of Jaffa Cakesand Kit Kats, and they chose Jaffa Cakes, we would write

C ({kitkat, jaffacakes}) = {jaffacakes}

• C is just a record of the choices made from all possible choicesets

• i.e. all sets in 2X apart from the empty set ∅

• We insist that the DM chooses something that was actually inthe data set

• i.e. C (A) ⊂ A

Notes

• Note that there is something a bit weird going on• We allow for people to choose more than one option!• i.e. we allow for data of the form

C ({kitkat, jaffacakes, lays}) = {jaffacakes, kitkat}

• Which we interpret as something like “the decision makerwould be happy with either jaffa cakes or lays from this choiceset”

• This is very useful, but a bit dubious• We will come back to it later

Utility Maximization

• The model we want to test is that of utility maximization• i.e. there exists a utility function u : X → R

• Such that the things that are chosen are those whichmaximize utility

• For every AC (A) = argmax

x∈Au(x)

• If this is true, we say that u rationalizes C• If C can be rationalized by some u then we say it has a utilityrepresentation

Representation Theorem

• We want to know when data is consistent with utilitymaximization

• i.e. it has a utility representation

• So we would like to find a set of conditions on C such that ithas a utility representation if and only if these conditions aresatisfied

• Testing these conditions is then the same as testing the modelof utility maximization


• You may remember a condition called the Weak Axiom ofRevealed Preference from Intermediate Micro

• We will break WARP down into two parts

Axiom α (AKA Independence of Irrelevant Alternatives) Ifx ∈ B ⊆ A and x ∈ C (A), then x ∈ C (B)

Axiom β If x , y ∈ C (A), A ⊆ B and y ∈ C (B) then x ∈ C (B)

• Notice we can test these conditions!• If we have data, we can see if they are satisfied


• These conditions form the basis of our first representationtheorem

TheoremA Choice Correspondence has a utility representation if and only ifit satisfies axioms α and β

• if: if α and β are satisfied then a utility representation exists

• only if: if a utility representation exists then α and β aresatisfied


• Because it is useful (and good for you) we are going to provethis (!)

• In order to do so, we are going to have to introduce anothermodel based on preferences• Again, should be familiar from Intermediate Micro

• Reminder: A preference relation � is a way of comparingalternatives

• If x is ‘as good as’y we write x � y• We write x � y if x � y but not y � x• We write x ∼ y if � y and y � x

Preference Relations

• We demand that preferences have certain properties:• Completeness: for every x and y in X either x � y or y � x(or both)

• Transitivity: if x � y and y � z then x � z• Reflexive: x � x

• We say that preference relation � represents a choice functionC if, for every A

C (A) = {x ∈ A|x � y for all y ∈ A}

• i.e. the things that are chosen are those that are preferred toeverything else in the choice set

Preferences and Utility

• Preferences are all well and good, but we were interested inthe model of utility maximization!

• How can we relate the two?• We say that a utility function u represents preferences � if

u(x) ≥ u(y) if and only if

x � y


• So if we can find• A preference relation which represents choices• A utility function which represents preferences

we are done!

• Preferences represents choices means

C (A) = {x ∈ A|x � y for all y ∈ A}

• Utility represents preferences means

u(x) ≥ u(y)⇐⇒ x � y

• So

C (A) = {x ∈ A|u(x) ≥ u(y) for all y ∈ A}= argmax

x∈Au(x)


• Thus, in order to prove that axioms α and β are equivalent toutility maximization we will do the following

1 Show that if the data satisfies α and β then we can find acomplete, transitive, reflexive preference relation � whichrepresents the data

2 Show that if the preferences are complete, transitive andreflexive then we can find a utility function u which representsthem

3 Show that if the data has a utility representation then it mustsatisfy α and β

• We will do 1 and 2 in class. You can do 3 for homework

From Choice to Preferences

• Our job is to show that, if choices satisfy α and β then we canfind a preference relation � which is• Complete, transitive and reflexive• Represents choices

TheoremA Choice Correspondence can be represented by a complete,transitive, reflexive preference relation if satisfies axioms α and β

From Choice to Preferences

• How should we proceed?1 Choose a candidate binary relation D2 Show that it is complete, transitive and reflexive3 Show that it represents choice

Guessing the Preference Relation

• If we observed choices, what do we think might tell us that xis preferred to y?

• How about if x is chosen when the only option is y?• Let’s try that!• We will define D as saying

x D y if x ∈ C (x , y)

• Okay, great, we have defined D• But we need it to have the right properties

Completeness

• Is D complete?• Yes!• For any set {x , y} either x or y must be chosen (or both)• In the former case x D y• In the latter y D x

Reflexiveness

• Is D reflexive?• Yes! (though we have been a bit cheeky)• Let x = y , so then C (x , x) = C (x) = x• Implies x D x

Transitivity

• Is D transitive?• Yes! (though this requires a little proving)• Assume not, then

x D y , y D zbut not x D z

• We need to show that this cannot happen• i.e. it violates α or β

• These are conditions on the data, so what do we need to do?• Understand what this means for the data

Transitivity

• Translating to the data• x D y means that x ∈ C (x , y)• y D z means that y ∈ C (y , z)• not x D z means that x /∈ C (x , z)

• Claim: such data cannot be consistent with α and β

• Why not?

Transitivity

• What would the person choose from {x , y , z}• x?

• No! Violation of α as x not chosen from {x , z}• y?

• No! This would imply (by α) that y ∈ C (x , y)• By β this means that x ∈ C (x , y , z)• Already shown that this can’t happen

• z?• No! This would imply (by α) that z ∈ C (y , z)• By β this means that y ∈ C (x , y , z)• Already shown that this can’t happen

Transitivity

• If we have x D y , y D z but not x D z then the data cannotsatisfy α and β

• Thus if α and β are satisfied, we know that D must betransitive!

• Thus, we can conclude that, if α and β are satisfied D musthave all three right properties!

Representing Choices

• Finally, we need to show that D represents choices - i.e.

C (A) = {x ∈ A|x D y for all y ∈ A}

• How do we do this?• Well, first note that we are trying to show that two sets areequal

• The set of things that are chosen• The set of things that are best according to D

• We do this by showing two things1 That if x is in C (A) it must also be x D y for all y ∈ A2 That if x D y for all y ∈ A then x is in C (A)

Things that are Chosen must be Preferred

• Say that x ∈ C (A)• For D to represent choices it must be that x D y for everyy ∈ A

• Note that, if y ∈ A, {x , y} ⊂ A• So by α if

x ∈ C (A)

⇒ x ∈ C (x , y)

• And so, by definitionx D y

Things that are Preferred must be Chosen

• Say that x ∈ A and x D y for every y ∈ A• Can it be that x /∈ C (A)• No! Take any y ∈ C (A)• By α, y ∈ C (x , y)• As x D y it must be the case that x ∈ C (x , y)• So, by β, x ∈ C (A)• Contradiction!

Done!

From Preference To Utility

• Well, unfortunately we are not really done• We wanted to test the model of utility maximization• So far we have shown that α and β are equivalent topreference maximization

• Need to show that preference maximization is the same asutility maximization

TheoremIf a preference relation % is complete, transitive and reflexive thenthere exists a utility function u : X → R which represents %, i.e.

u(x) ≥ u(y)⇐⇒ x � y


• I am going to sketch the proof because you might find itinteresting

• However, I won’t ask you to reproduce this on an exam, soyou can relax if you so wish

Proof By Induction

• We are going to proceed using proof by induction• We want to show that our statement is true regardless of thesize of X

• We do this using induction on the size of the set• Let n = |X |, the size of the set

• Induction works in two stages• Show that the statement is true if n = 1• Show that, if it is true for n, it must also be true for any n+ 1

• This allows us to conclude that it is true for n• It is true for n = 1• If it is true for n = 1 it is true for n = 2• If it is true for n = 2, it is true for n = 3....

• You have to be a bit careful with proof by induction• Or you can prove that all the horses in the world are the samecolor


• So in this case we have to show that we can find a utilityrepresentation if |X | = 1• Trivial

• And show that if a utility representation exists for |X | = n,then it exists for |X | = n+ 1• Not trivial

Step 1

• Take a set such that |X | = n+ 1 and a complete, transitivereflexive preference relation �

• Remove some x∗ ∈ X• Note that the new set X/x∗ has size n• So, by the inductive assumption, there exists somev : X/x∗ → R such that

v(x) ≥ v(y)⇐⇒ x � y

• So now all we need to do is assign a utility number to x∗which makes it work with v

• How would you do this?

Step 2

• Four possibilities1 x∗ ∼ y for some y ∈ X/x∗

• Set v (x∗) = v (y )

2 x∗ % y for all y ∈ X/x∗

• Set v (x∗) = maxy∈X /x ∗ v (y ) + 1

3 x∗ � y for all y ∈ X/x∗

• Set v (x∗) = miny∈X /x ∗ v (y )− 1

4 None of the above

Step 2

• What do we do in case 4?• We divide X in two: those objects better than x∗ and thoseworse than x∗

X∗ = {y ∈ X/x∗|x∗ � x}

X ∗ = {y ∈ X/x∗|x � x∗}

• Figure out the highest utility in X∗ and the lowest utility inX ∗ and fit the utility of x∗ in between them

v(x∗) =12miny∈X ∗

v(y) +12maxy∈X∗

v(y)

Step 2

• Note that everything in X ∗ has higher utility than everythingin X∗• Pick an x ∈ X ∗ and y ∈ X∗• x � x∗ and x∗ � y• Implies x � y (why?)• and so v(x) ≥ v(y)• In fact, because we have ruled out indifference v(x) > v(y)

• This implies that

v(x) > v(x∗) > v(y)

• And so• The utility of everything better than x∗ is higher than v(x∗)• The utility of everything worse than x∗ is lower than v(x∗)

Step 3

• Verify that v represents � in all of the four cases• That sounds exhausting• You can look in the lecture notes if you so wish

Done!


• The final step is to show that, if a choice correspondence hasa utility representation then it satisfies α and β

• This closes the loop and shows that all the statements areequivalent

• A choice correspondence satisfies α and β• A choice correspondence has a preference relation• A choice correspondence has a utility representation

• Will leave you to do that for homework!


• We have now achieved our aim!• We know how to test the model of utility maximization

TheoremA Choice Correspondence has a utility representation if and only ifit satisfies axioms α and β

• We just test α and β

• Before we move on to something more fun, I want to discusstwo potential issues

• How seriously should we take utility?• What happens if our data is not as good as we would like it tobe?

How Unique is Our Utility Function?

• We now know that if α and β are satisfied, we can find someutility function that will explain choices

• Is it the only one?


Croft’s ChoicesAvailable Snacks Chosen SnackJaffa Cakes, Kit Kat Jaffa CakesKit Kat, Lays Kit KatLays, Jaffa Cakes Jaffa CakesKit Kat, Jaffa Cakes, Lays Jaffa Cakes

• These choices could be explained by u(J) = 3, u(K ) = 2,u(L) = 1

• What about u(J) = 100000, u(K ) = −1, u(L) = −2?• Or u(J) = 1, u(K ) = 0.9999, u(L) = 0.998?


• In fact, if a data set obeys α and β there will be many utilityfunctions which will rationalize the data

TheoremLet u : X → R be a utility representation for a ChoiceCorrespondence C. Then v : X → R will also represent C if andonly if there is a strictly increasing function T such that

v(x) = T (u(x)) ∀ x ∈ X

• Strictly increasing function means that if you plug in a biggernumber you get a bigger number out


Snack u v w

Jaffa Cake 3 100 4Kit Kat 2 10 2Lays 1 −100 3

• v is a strictly increasing transform on u, and so represents thesame choices

• w is not, and so doesn’t• For example think of the choice set {k, l}• u says they should choose kit cat• w says they should choose lays

Why Does This Matter?

• It is important that we know how much the data can tell usabout utility

• Or other model objects we may come up with

• For example, our results tell us that there is a point indesigning a test to tell whether people maximize utility

• But there is no point in designing a test to see whether theutility of Kit Kats is twice that of Lays• Assuming α and β is satisfied, we can always find a utilityfunction for which this is true

• And another one for which this is false!

• We can use choices to help us determine that the utility of KitKats is higher than the utility of Lays

• But nothing in our data tells us how much higher is theutility of Kit Kats

Choices from all Choice Sets?

• Imagine running an experiment to try and test α and β

• The data that we need is the choice correspondence

C : 2X /∅→ 2X /∅

• How many choices would we have to observe?• Lets say |X | = 10

• Need to observe choices from every A ∈ 2X /∅• How big is the power set of X ?• If |X | = 10 need to observe 1024 choices• If |X | = 20 need to observe 1048576 choices

• This is not going to work!

Choices from all Choice Sets?

• So how about we forget about the requirement that weobserve choices from all choice sets

• Are α and β still enough to guarantee a utility representation?

C ({x , y}) = x

C ({y , z}) = y

C ({x , z}) = z

• If this is our only data then there is no violation of α or β

• But no utility representation exists!• We need a different approach!

Revealed Preference

• We say that x is directly revealed preferred to y if, forsome choice set A

y ∈ A

x ∈ C (A)

• We say that x is revealed preferred to y if we can find a setof alternatives w1, w2, ....wn such that• x is directly revealed preferred to w1• w1 is directly revealed preferred to w2• ...• wn−1 is directly revealed preferred to wn• wn is directly revealed preferred to y

• We say x is strictly revealed preferred to y if, for somechoice set A

y ∈ A but not y ∈ C (A)x ∈ C (A)

The Generalized Axiom of Revealed Preference

• Note that we can observe revealed preference and strictrevealed preference from the data

• With these definitions we can write an axiom to replace α andβ

DefinitionA choice correspondence C satisfies the Generalized Axiom ofRevealed Preference (GARP) if it is never the case that x isrevealed preferred to y , and y is strictly revealed preferred to x

TheoremA choice correspondence C satisfies GARP if and only if it has autility representation. This is true even if C is incomplete (i.e. doesnot report choices from all choice sets)

Choice Correspondence?

• Another weird thing about our data is that we assumed wecould observe a choice correspondence

• This is not an easy thing to do!• What about if we only get to observe a choice function?• How do we deal with indifference?

Choice Correspondence?

• One of the things we could do is assume that the decisionmaker chooses one of the best options

C (A) ∈ argmaxx∈A

u(x)

• Is this going to work?• No!• Any data set can be represented by this model

• Why?• We can just assume that all alternatives have the same utility!

• Need some way of identifying when an alternative x is betterthan alternative y• i.e. some way to identify strict preference

Choice from Budget Sets

• One case in which we can do this is if our data comes frompeople choosing consumption bundles from budget sets• Should be familiar from intermediate micro

• The objects that the DM has to choose between are bundlesof different commodities

x =

x1...xn

• And they can choose any bundle which satisfies their budgetconstraint {

x ∈ Rn+|

n

∑i=1pixi ≤ I

}

Choice from Budget Sets

Monotonicity

• Claim: We can use choice from budget sets to identify strictpreference

• Even if we only see a single bundle chosen from each budgetset

• As long as we assume more is better

xn ≥ yn for all n and xn > yn for some n

implies that x � y

• i.e. preferences are strictly monotonic

Monotonicity

Monotonicity

• Claim: if px is the prices at which the bundle x was chosen

pxx > pxy implies x � y

• Why?

Revealed Strictly Preferred

• Because x was chosen, we know x % y• Because pxx > pxy we know that y was inside the budgetset when x was chosen

• Could it be that y % x?

Revealed Strictly Preferred

• Because y is inside the budget set, there is a z which is betterthan y and affordable when x was chosen

• Implies that x % z and (by monotonicity) z � y• By transitivity x � y

Revealed Preference

• When dealing with choice from budget sets we say

• x is directly revealed preferred to y if px x ≥ px y• x is revealed preferred to y if we can find a set ofalternatives w1, w2, ....wn such that

• x is directly revealed preferred to w1• w1 is directly revealed preferred to w2• ...• wn−1 is directly revealed preferred to wn• wn is directly revealed preferred to y

• x is strictly revealed preferred to y if px x > px y

Afriat’s Theorem

Theorem (Afriat)Let {x1, .....x l} be a set of chosen commodity bundles at prices{p1, ..., pl

}. The following statements are equivalent:

1 The data set can be rationalized by a strictly monotonic set ofpreferences � that can be represented by a utility function

2 The data set satisfies GARP

3 There exists a continuous, concave, piecewise linear, strictlymonotonic utility function u that rationalizes the data

Summary

• We have completed our review of what is sometimes called’revealed preference theory’

• Phew

• Here are the takeaways

Summary

1 Testing models which have unobserved (latent) variables istricky• For example the model of utility maximization

2 The gold standard is a ’representation theorem’• Conditions on the data which are equivalent to testing themodel

• Don’t have to make any specific assumptions about the natureof utility

3 In the case of utility maximization, we have such conditions• α and β

4 These work if we can observe choice correspondences fromevery choice set• Otherwise we need to use GARP

5 The utility numbers we find are not unique• Only tell us ordering, not magnitudes

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A Representation Theorem for Utility Maximization